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IEEE International Symposium on Multiple-Valued Logic (ISMVL)

Fourier decision diagrams on finite non-Abelian groups with preprocessing

Ratko Stanković

Published Nov 22, 2002• 13 citations
Publisher

Abstract

This paper introduces a method for optimization of Fourier decision diagrams representations of discrete functions based on the preprocessing of the represented functions. Fourier decision diagrams are defined for matrix-valued-function on non-Abelian groups. Their application to number-valued functions is provided through a simple preprocessing of the represented functions.

DOI
10.1109/ismvl.1997.601415

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This paper introduces a method for optimization of Fourier decision diagrams representations of discrete functions based on the preprocessing of the represented functions. Fourier decision diagrams are defined for matrix-valued-function on non-Abelian groups. Their application to number-valued functions is provided through a simple preprocessing of the represented functions.
Spectral techniques on Abelian groups are a well-established tool in diverse fields such as signal processing, switching theory, multi-valued logic and logic design. The harmonic analysis on finite non-Abelian groups is an extension of them, which has also found applications for particular tasks in the same fields. It takes advantages of the peculiar features of the domain groups and their dual objects. Representing unitary irreducible representations, that are kernels of Fourier transforms on non-Abelian groups, in a compact manner is a key task in this area. These representations are usually specified in terms of rectangular matrices with matrix entries. Therefore, the problem of their efficient representations can be viewed as handling large rectangular matrices with matrix-valued entries. Quantum Multiple-valued Decision Diagrams (QMDDs) and Heterogeneous Decision Diagrams (HDDs) have been used for representation of matrices with numerical values, under some restrictions to the order of matrices to be represented. In this paper, we present a generalization of this concept for the representation of rectangular matrices with matrix-valued entries. We also demonstrate an implementation of an XML-based software package aimed at handling such data structures.
Quantum Fourier transformations are an essential component of many quantum algorithms, from prime factoring to quantum simulation. While the standard Abelian QFrT is well studied, important variants corresponding to non-Abelian groups of interest have seen less development. In particular, fast non-Abelian Fourier transformations are important components for both quantum simulations of field theories as well as approaches to the non-Abelian hidden subgroup problem. In this work, we present fast quantum Fourier transformations for a number of non-Abelian groups of interest for high energy physics, BT\mathbb{B}\mathbb{T}BT, BO\mathbb{B}\mathbb{O}BO, $6\mathrm{\ensuremath{\Delta}}(27)$, $\mathrm{\ensuremath{\Delta}}(54)$, and $\mathrm{\ensuremath{\Sigma}}(36\ifmmode\times\else\texttimes\fi{}3)$. For each group, we derive explicit quantum circuits and estimate resource scaling for fault-tolerant implementations. Our work shows that the development of a fast Fourier transformation can substantively reduce simulation costs by an up to three orders of magnitude for the finite groups that we have investigated.
This paper presents a decision diagram structure intended for the simulation and verification of reversible and quantum circuits. The structure is designed to efficiently represent the matrices describing reversible and quantum gate and circuit behaviour and takes advantage of key properties of those matrices. Algorithms are described for efficiently building the decision diagram representation directly from a gate without constructing the actual matrix and for performing matrix multiplication which is the fundamental operation to determine the function realized by a cascade of gates. Experimental results show the methods presented are applicable to large circuits at the state of the art in reversible and quantum synthesis and design.
The problem of synthesis and optimisation of logical reversible and quantum circuits from functional descriptions represented as decision diagrams is considered. It is one of the key problems being solved with the aim of creating quantum computing technology and quantum computers. A new method of stepwise transformation of the initial functional specification to a quantum circuit is proposed, which provides for the following project states: reduced ordered binary decision diagram, if-decision diagram, functional if-decision diagram, reversible circuit and quantum circuit. The novelty of the method consists in extending the Shannon and Davio expansions of a Boolean function on a single variable to the expansions of the same Boolean function on another function with obtaining decomposition products that are represented by incompletely defined Boolean functions. Uncertainty in the decomposition products gives remarkable opportunities for minimising the graph representation of the specified function. Instead of two outgoing branches of the binary diagram vertex, three outgoing branches of the if-diagram vertex are generated, which increase the level of parallelism in reversible and quantum circuits. For each transformation step, appropriate mapping rules are proposed that reduce the number of lines, gates and the depth of the reversible and quantum circuit. The comparison of new results with the results given by the known method of mapping the vertices of binary decision diagram into cascades of reversible and quantum gates shows a significant improvement in the quality of quantum circuits that are synthesised by the proposed method.
Hoyer has given a generalisation of the Deutsch--Jozsa algorithm which uses the Fourier transform on a group G which is (in general) non-Abelian. His algorithm distinguishes between functions which are either perfectly balanced (m-to-one) or constant, with certainty, and using a single quantum query. Here, we show that this algorithm (which we call the Deutsch--Jozsa--Hoyer algorithm) can in fact deal with a broader range of promises, which we define in terms of the irreducible representations of G.
In recent years the use of decision diagrams within the context of discrete optimization has proliferated. This paper continues this expansion by proposing the use of decision diagrams for modeling and solving binary optimization problems with quadratic constraints. The model proposes the use of multiple decision diagrams to decompose a quadratic matrix so that each individual diagram has provably limited size. The decision diagrams are then linked through channeling constraints to ensure that the solution represented is consistent across the decision diagrams and that the original quadratic constraints are satisfied. The resulting family of decision diagrams are optimized over by a dedicated cutting-plane algorithm akin to Benders decomposition. The approach is general, in that commercial integer programming solvers can readily apply the technique. A thorough experimental evaluation on both benchmark and synthetic instances exhibits that the proposed decision diagram reformulation provides significant improvements over current methods for quadratic constraints in state-of-the-art solvers.